Please Help guys. Jupiter\'s Cosmic Slingshot: lo\'s Plasma torus Jupiter\'s moo

Question

Please Help guys.

Jupiter's Cosmic Slingshot: lo's Plasma torus Jupiter's moon lo is the most volcanically active body in the solar system. Because it orbits inside Jupiter's powerful magnetic field, a large amount of the material in lo's volcanic plumes is ionized and trapped by Jupiter s swiftly rotating magnetic field. This creates a plasma torus of ions around Jupiter. Data o's orbital period 75 days 1.51 x 105 s. mass of Jupiter 1.90 x 107 kg. radius of lo's orbit 4.22 x 108 m Period of Jupiters magnetic field rotation 10 hours speed of ions at lo's distance from Jupiter 7.4 x 104 m/s G 6.67 x 10 11 Nmaikg a) Calculate the velocity for a particle in orbit around Jupiter at the same distance as lo by setting the gravitational force equal to the centripetal force. b) Calculate the period for the ions moving at 7.4 x 1 m/s at the same distance from Jupiter as lo. What does the answer say about what is likely keeping the ions moving at this speed? c) Calculate the angular momentum per kg for the ions in part b. d) over time, many of these ions regain an electron, becoming neutral. They then have too large a velocity for gravity to hold them in an orbit as tight as lo's. The angular momentum is conserved. Write the expression for the velocity in terms of the angular momentum

Explanation / Answer

a)
m*v^2/r = G*M*m/r^2

==> v = sqrt(G*M/r)

= sqrt(6.67*10^-11*1.9*10^27/4.22*10^8)

= 1.73*10^4 m/s

= 17.3 km/s

b) T = 2*pi*r/v

= 2*pi*4.22*10^8/1.73*10^4

= 1.532*10^5 s

= 42.56 hours

c) L = m*v*r

d) ==> v = L/(m*r)

e) m*v^2/r = G*M*m/r^2

r = G*M/v^2

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